Sum-product exponent for the reals
Description of constant
For a finite set $A \subset \mathbb{R}$ write \(A+A = \\{ a+b : a,b \in A \\}, \qquad AA = \\{ ab : a,b \in A \\}\) for the sumset and product set. The (real) sum-product exponent is \(C_{84b} := \liminf_{n \to \infty}\ \min_{\substack{A \subset \mathbb{R} \\ \lvert A\rvert = n}} \frac{\log \max(\lvert A+A\rvert, \lvert AA\rvert)}{\log n}.\) Equivalently, $C_{84b}$ is the largest $s$ such that $\max(\lvert A+A\rvert, \lvert AA\rvert) \gg_\epsilon \lvert A\rvert^{s-\epsilon}$ for every $\epsilon > 0$ and every finite $A \subset \mathbb{R}$.
Erdős and Szemerédi [ErSz83] conjectured (originally for $A \subset \mathbb{Z}$, but also for $A \subset \mathbb{R}$) that $C_{84b} = 2$: at least one of the sumset or product set must be of near-maximal size $\lvert A\rvert^{2-o(1)}$. This sum-product conjecture for the reals was disproved in May 2026 by Bloom, Sawin, Schildkraut and Zhelezov [BSSZ2026], who showed $C_{84b} < 2$.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $2$ | [ErSz83] | Erdős–Szemerédi constructed sets of integers with $\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{2-c/\log\log\lvert A\rvert}$, so the exponent is $\le 2$. This was conjectured to be sharp (the sum-product conjecture). |
| $2 - c$ for an absolute $c>0$ | [BSSZ2026] | Bloom–Sawin–Schildkraut–Zhelezov disprove the conjecture by constructing arbitrarily large $A \subset \mathbb{R}$ — algebraic integers in totally real number fields of degree $\asymp \log\lvert A\rvert$ — with $\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{2-c}$. A non-optimized explicit version of the argument gives $c \ge 0.00000087$, i.e. exponent $\le 1.99999913$ (the authors stress this value “should not be taken too seriously”). |
| $1.999281$ | [Al26], [EPF52] | Althoefer (28 May 2026), posted on the Erdős Problems forum [EPF52]; a ChatGPT 5.5 long-thinking optimization of the explicit constant of [BSSZ2026, §5] giving $c \ge 0.000719$. Unverified. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial | $\lvert A+A\rvert \ge 2\lvert A\rvert - 1$. |
| $5/4 = 1.25$ | [El97] | Elekes, via the Szemerédi–Trotter incidence theorem. |
| $14/11 \approx 1.2727$ | [So05] | Solymosi. |
| $4/3 \approx 1.33333$ | [So09] | Solymosi; the long-standing “$4/3$ barrier”. |
| $\approx 1.33338$ | [KoSh16] | Konyagin–Shkredov; first to break the $4/3$ barrier (their 2015 and 2016 papers give $\approx 1.33338$ and $\approx 1.33384$). |
| $\approx 1.33428$ | [Sh19] | Shakan. |
| $\tfrac{1558}{1167} \approx 1.33504$ | [RuSt22] | Rudnev–Stevens. |
| $\tfrac{1270}{951} \approx 1.33543$ | [Bl25] | Bloom. |
| $\tfrac{4}{3} + \tfrac{10}{4407} \approx 1.335602$ | [Cu25] | Cushman; current record. |
Additional comments and links
- The sum-product conjecture is studied over many other rings — $\mathbb{Z}$, $\mathbb{C}$, the quaternions, finite fields $\mathbb{F}_p$ and $\mathbb{F}_q$, $p$-adics $\mathbb{Q}_p$, function fields, and matrix rings — each with its own chronology of bounds. To avoid clutter we do not reproduce these here; Thomas Bloom maintains a comprehensive and up-to-date history of all variants at [Bloom-notes]. It is a notorious open problem whether $C_{84b}$ equals the corresponding integer exponent $c(\mathbb{Z})$.
- The same paper [BSSZ2026] also disproves the many sums and products conjecture of Erdős (constructing, for each $k \ge 3$, arbitrarily large $A \subset \mathbb{R}$ with $\max(\lvert kA\rvert, \lvert A^{(k)}\rvert) \le \lvert A\rvert^{C \log k/\log\log k}$), gives new lower bounds for the number of solutions to linear equations in multiplicative groups and to the unit equation, and produces analogous counterexamples over $\mathbb{Q}_p$, $\mathbb{F}_p$, and function fields. In small characteristic the function-field exponents are much smaller — e.g. for $q = 1024$ one gets $\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{1.906}$.
- The disproof was directly inspired by the OpenAI counterexample to the Erdős unit distance conjecture (84a); both rely on totally real (or split) number fields of large degree with discriminant $O(1)^d$, coming from Martinet’s class field towers and a Golod–Shafarevich argument. The authors note that, surprisingly, the sum-product construction required less number theory than the unit distance one (in particular it does not need small split primes).
- The analogous problem for finite sets of integers $A \subset \mathbb{Z}$ — Erdős Problem #52, [EP52] — is technically still open: although [BSSZ2026] disproves the conjecture for $A \subset \mathbb{R}$, the construction uses algebraic integers in number fields of degree growing with $\lvert A\rvert$, and the integer case is unaffected. The best known integer lower bound is still Cushman’s $\lvert A\rvert^{4/3 + 10/4407}$ [Cu25].
References
- [ErSz83] Erdős, Paul; Szemerédi, Endre. On sums and products of integers. In: Studies in Pure Mathematics, 213–218, Birkhäuser, Basel, 1983.
- [El97] Elekes, György. On the number of sums and products. Acta Arithmetica 81 (1997), no. 4, 365–367.
- [So05] Solymosi, József. On the number of sums and products. Bulletin of the London Mathematical Society 37 (2005), no. 4, 491–494.
- [So09] Solymosi, József. Bounding multiplicative energy by the sumset. Advances in Mathematics 222 (2009), no. 2, 402–408.
- [KoSh16] Konyagin, Sergei V.; Shkredov, Ilya D. New results on sums and products in $\mathbb{R}$. Trudy Mat. Inst. Steklova 294 (2016), 87–98; translation in Proc. Steklov Inst. Math. 294 (2016), 78–88.
- [Sh19] Shakan, George. On higher energy decompositions and the sum-product phenomenon. Mathematical Proceedings of the Cambridge Philosophical Society 167 (2019), no. 3, 599–617.
- [RuSt22] Rudnev, Misha; Stevens, Sophie. An update on the sum-product problem. Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411–430. arXiv:2005.11145.
- [Bl25] Bloom, Thomas F. Sum-product estimate over the reals (2025). See [Bloom-notes] for the precise reference.
- [Cu25] Cushman, Adam. A note on the sum-product problem and the convex sumset problem. arXiv:2512.13849 (2025).
- [BSSZ2026] Bloom, Thomas F.; Sawin, Will; Schildkraut, Carl; Zhelezov, Dmitrii. The sum-product conjecture is false for real numbers. arXiv:2605.28781 (2026).
- [Bloom-notes] Bloom, Thomas F. A history of the sum-product problem. thomasbloom.org/notes/sumproduct.html (accessed 28 May 2026).
- [Al26] Althoefer, Ingo. Improved constant for [BSSZ2026]. Note (28 May 2026), with tex and pdf.
- [EPF52] Bloom, Thomas F. (ed.). Erdős Problem #52: Discussion thread. erdosproblems.com/forum/thread/52 (accessed 29 May 2026).
- [EP52] Bloom, Thomas F. Erdős Problem #52 (the integer sum-product conjecture, still open). erdosproblems.com/52 (accessed 29 May 2026).
Contribution notes
Prepared with assistance from Claude Opus 4.7, which read the [BSSZ2026] preprint and Bloom’s history page to assemble the bounds. The pre-2026 lower-bound citations were compiled from Bloom’s history page together with the reference list of [BSSZ2026], and the exact published details (volumes, page numbers, the [Bl25] title) should be independently verified before citation.